Question

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Find an equation of the tangent line to the function
y = 3x2
at the point P(1, 3).
Solution
We will be able to find an equation of the tangent line ℓ as soon as we know its slope m. The difficulty is that we know only one point, P, on ℓ, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point
Q(x, 3x2)
on the parabola (as in the figure below) and computing the slope mPQ of the secant line PQ. [A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.]

Answer 1

The equation of the tangent line to the quadratic function y = 3 · x² at the point (x, y) = (1, 3) is y = 6 · x - 3.

How to determine the equation of a line tangent to a quadratic equation by algebraic methods

Herein we must determine a line tangent to the quadratic equation y = 3 · x² at the point P(x, y) = (1, 3) by algebraic means. The slope of the line can be found by using the secant line formula and simplify the resulting expression:

m = [3 · (x + Δx)² - 3 · x²] / [(x + Δx) - x]

m = 3 · [(x + Δx)² - x²] / Δx

m = 3 · (x² + 2 · x · Δx + Δx ² -  x²) / Δx

m = 3 · (2 · x + Δ x)

If Δx = 0, then the equation of the slope of the tangent line is:

m = 6 · x

If we know that x = 1, then the slope of the tangent line is:

m = 6 · 1

m = 6

Lastly, we find the intercept of the equation of the line: (x, y) = (1, 3), m = 6

b = y - m · x

b = 3 - 6 · 1

b = - 3

The equation of the tangent line to the quadratic function y = 3 · x² at the point (x, y) = (1, 3) is y = 6 · x - 3.

To learn more on tangent lines: https://brainly.com/question/23265136

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